In other words, Hilberts 10. Problem is undecided.
The mathematicians hoped to follow the same approach to prove the extended version of the problem-but they met a catch.
Build up the work
The useful correspondence between Turing machines and diophantine equations falls apart if the equations may have non-insulating solutions. For example, consider the equation again y = X2. If you work in a ring of whole numbers that contain √2, you will receive some new solutions, such as: X = √2, y = 2. The equation no longer corresponds to a Turing machine that calculates the perfect squares – and in general the diophantine equations can no longer codes the institutional problem.
But in 1988 a doctoral student was named at New York University Sasha Shlapentokh started playing with ideas how to deal with this problem. Until 2000, they and others had formulated a plan. Say you should add an equation like a few additional terms y = X2 That magically forced it X Be an integer again, even in a different number system. Then you could save the correspondence to a Turing machine. Could the same be done for all diophantine equations? In this case, this would mean that Hilbert’s problem could encodes the stirring problem in the new number system.
Illustration: Myriam were for How much magazine
Over the years, Shlapentokh and other mathematicians found out which terms they had to contribute to the Diophantine equations for different types of rings, which made it possible for them to demonstrate that Hilbert’s problem was still undecided in these environments. Then they cooked all the remaining rings down to a case: rings that affect the imaginary number I. Mathematicians recognized that in this case the terms they would have to add are determined by a special equation called an elliptical curve.
But the elliptical curve would have to meet two properties. First, it should have an infinite number of solutions. Second, all solutions for the elliptical curve would have to switch to another ring from full – if you have removed the imaginary number from your number system, maintain the same underlying structure.
As it turned out, it was an extremely subtle and difficult task to build such an elliptical curve that worked for every remaining ring. But Komans and Pagano – exceed elliptical curves that had worked closely together since their graduate school – had exactly the right tool that should try it out.
Sleepless nights
Since his time as a student, Koymans had thought about Hilbert’s 10th problem. Throughout his entire graduate school, it waved during his collaboration with Pagano. “Every year I spent a few days about thinking about it and capturing it terribly,” said Komans. “I would try three things and they would all hunt my face in the air.”
In 2022 he talked to the problem at a conference in Banff, Canada, and Pagano. They hoped that they were able to build the special elliptical curve that is necessary to solve the problem. After finishing some other projects, they got to work.
